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1
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Invariant measures for nonexpansive Markov operators on Polish spaces

100%
Szarek T.
Instytut Matematyczny Polskiej Akademii Nauk
EN
New sufficient conditions for the existence of an invariant measure for nonexpansive Markov operators defined on Polish spaces are presented. These criteria are applied to iterated function systems, stochastically perturbed dynamical systems and Poisson stochastic differential equations. We also estimate the Ledrappier version of capacity for invariant measures.
2
Content available remote

The stability of Markov operators on Polish spaces

95%
Szarek T.
Studia Mathematica
|
2000
|
tom 143
|
nr 2
145-152
EN
A sufficient condition for the asymptotic stability of Markov operators acting on measures defined on Polish spaces is presented.
3
Content available remote

Invariant measures for iterated function systems

95%
Szarek T.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 1
87-98
EN
A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.
4
Content available remote

Markov operators acting on Polish spaces

95%
Szarek T.
Annales Polonici Mathematici
|
1997
|
tom 67
|
nr 3
247-257
EN
We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.
5
Content available remote

Generic properties of learning systems

95%
Szarek T.
Annales Polonici Mathematici
|
2000
|
tom 73
|
nr 2
93-103
EN
It is shown that the set of learning systems having a singular stationary distribution is generic in the family of all systems satisfying the average contractivity condition.
6
Content available remote

The uniqueness of invariant measures for Markov operators

95%
Szarek T.
Studia Mathematica
|
2008
|
tom 189
|
nr 3
225-233
EN
It is shown that Markov operators with equicontinuous dual operators which overlap supports have at most one invariant measure. In this way we extend the well known result proved for Markov operators with the strong Feller property by R. Z. Khas'minski.
7
Content available remote

Invariant measures for Markov operators with application to function systems

95%
Szarek T.
Studia Mathematica
|
2003
|
tom 154
|
nr 3
207-222
EN
A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems.
8
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Lower bound technique and its applications to function systems and stochastic partial differential equations

95%
Szarek T.
Mathematica Applicanda
|
2013
|
tom 41
|
nr 2
PL
W pracy formułujemy kryteria dla istnienia miary niezmienniczej dla łańcuchów i procesów Markowa. Następnie pokazujemy ich użyteczność w teorii iterowanych układów funkcyjnych i stochastycznych równań  różniczkowych.
EN
We formulate some criteria on the existence of an invariant measure for Markov chains and Markov processes. We also show their utylities in the theory of function systems and stochastic differential equations
9
Content available remote

Estimates of capacity of self-similar measures

60%
Myjak J., Szarek T.
Annales Polonici Mathematici
|
2002
|
tom 78
|
nr 2
141-157
EN
We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems ${(S_i,p_i): i=1,...,N}$ where $S_i$ are bi-lipschitzean transformations.
10
Content available remote

On the Hausdorff Dimension of Topological Subspaces

60%
Szarek T., Ślęczka M.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom 54
|
nr 3
201-205
EN
It is shown that every Polish space X with $dim_{T} X ≥ d$ admits a compact subspace Y such that $dim_{H} Y ≥ d$ where $dim_{T}$ and $dim_{H}$ denote the topological and Hausdorff dimensions, respectively.
11
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Irreducible Markov systems on Polish spaces

60%
Horbacz K., Szarek T.
Studia Mathematica
|
2006
|
tom 177
|
nr 3
285-295
EN
Contractive Markov systems on Polish spaces which arise from graph directed constructions of iterated function systems with place dependent probabilities are considered. It is shown that their stability may be studied using the concentrating methods developed by the second author [Dissert. Math. 415 (2003)]. In this way Werner's results obtained in a locally compact case [J. London Math. Soc. 71 (2005)] are extended to a noncompact setting.
12
Content available remote

Szpilrajn type theorem for concentration dimension

60%
Myjak J., Szarek T.
Fundamenta Mathematicae
|
2002
|
tom 172
|
nr 1
19-25
EN
Let X be a locally compact, separable metric space. We prove that $dim_{T} X = inf{dim_{L} X': X' is homeomorphic to X}$, where $dim_{L} X$ and $dim_{T} X$ stand for the concentration dimension and the topological dimension of X, respectively.
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